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Full text of "Treatise On Analysis Vol-Ii"

10 SPACES WITH OPERATORS. ORBIT SPACES 47 If E is a Banach space, the group GL(E) of linear homeomorphisms of E onto E (12.8.1) acts continuously on E by (u, x)i~+u(x) (5.7.4). Let G be a topological group and E a topological space. The mapping (s9 x)i-+x of G x E into E is a continuous action of G on E, called the trivial action. Let G be a topological group acting continuously on a topological space E. If p is a continuous homomorphism of a topological group G' into G, then G' acts continuously on E by (s'y x)\~*p(sf) - x. (12.10.3) If G is a topological group acting continuously on a topological space E, then for each s e G the mapping xt-*s - x is a homeomorphism of E onto E. For it is a continuous bijection, and the inverse bijection x^^s"1 • xis also continuous. (12.10.4) If G is a topological group acting continuously on a Hausdorff topological space E, the stabilizer of each point ofE is a closed subgroup ofG. This follows immediately from (12.3.5). (12.10.5) Let G be a metrizable group acting continuously on a metrizable space E. Let A be a compact subset of G, and B a closed (resp. compact) subset ofE. Then A • B is closed (resp. compact) in E. The second assertion follows from the fact that A • B is the image of the compact set A x B (3.20.1 6(v)) by the continuous mapping (s,x)h-*s-x (3.17.9). As to the first assertion, consider a sequence (sn • xn) of points of A • B (where sn e A and xn e B) with a limit z e E. By hypothesis, the se- quence (sn) has a subsequence (snk) converging to some point a e A. Since xnk = Snkl - (snk • xttk), the sequence (xnj) converges to a""1 • z. But B is closed in E and therefore a""1 • z e B, hence z = a - (a""1 • z) e A • B. By (3.13.13), the proof is complete. Let G be a topological group acting continuously on a topological space E. Let E/G be the set of orbits and let TC : E -> E/G be the canonical mapping, so that n(x) is the orbit G - x for each x e E. Let C be the set of subsets U of E/G such that TiT^U) is open in E. It follows immediately from the formulas (1.8.5) and (18,6) that D is a topology on E/G. The set E/G, endowed with G, the product group H x K acts continuously